8
1 Gamma Functions, Beta Functions, and Related
optional second one if the method should be applied on the property “value” or the
input value z “inz”. The default is “value”.
Example
r =0.6;
% z coordinates
theta = linspace(0,2 * pi,200);
z = r * exp(i * theta);
obj = psiC(z);
% psi-function
% method angle:
phi = obj.angle(’value’);
%
phase angle
rho = obj.abs(’value’);
% method abs: radius
polarplot(phi,rh)
The toolbox comes in addition with example psivisu.
1.5
The Incomplete Gamma Functions
1.5.1 Fundamental Equations
There are two incomplete gamma functions which differ in the selected integration
tail.
γ (z, a) =
z
0
exp(−t)t
a−1 dt, and
(1.12)
Γ (z, a) =
∞
z
exp(−t)t
a−1 dt.
(1.13)
Their sum is given by the Gamma function
Γ (a) = γ (x, a) + Γ (x, a),
(1.14)
thus it is sufficient to compute one type of the incomplete gamma functions. The
normalized functions are
P (z, a) =
γ (z, a)
Γ (a)
and Q(z, a) =
Γ (z, a)
Γ (a)
with
(1.15a)
P (z, a) +Q(z, a) = 1.
(1.15b)
γ (z, a) has simple poles at a equal to negative integers or zero.
The incomplete gamma functions hold the recurrence formula
γ (z, a + 1) = aγ (z, a) − z
a exp(−z) and
(1.16)